Averages of Green Functions of Classical Groups

概要

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In this paper, we compare the Green functions of $Sp(2n,q)$ and $SO(2n+1,q)$ with those of $GL(n,q^2)$ and find an interesting connection between them. Let $G = Sp_{2n}(\FFq)$ or $SO_{2n+1}(\FFq)$ and $\bar G = GL_n(\FFq)$ with Frobenuius map $F$. The Weyl group $W$ of $G$ is written as $W = DS_n$, where $D$ is an elementary abelian 2-group and $S_n$ is the symmetric group of degree $n$, which is identified with the Weyl group of $\bar G$. Let $Q_{T_w}^G$ be a Green function of $G^F$ where $T_w$ is an $F$-stable maximal torus of $G$ corresponding to $w \in W$. For $w \in S_n$, we define an average of Green functions $Q_{w, D}^G$ on $G^F$ by $Q_{w,D}^G = |D|\iv\sum_{x \in D}Q_{T_{wx}}^G.$ Then there exists a natural injection $u_0 \mapsto u$ from the set of unipotent classes of $\bar G$ to the set of unipotent classes of $G$ such that the function $Q_{w,D}^G(u)$ on $G^F$ coincides with the Green function $Q_{\bar T_w}^{\bar G}(u_0)$ on $\bar G^{F^2}$.
東京大学の論文